We consider the propagation of TE-polarized electromagnetic waves in cylindrical dielectric waveguides of circular cross section filled with lossless, nonmagnetic, and isotropic medium exhibiting a local Kerr-type dielectric nonlinearity. We look for axially-symmetric solutions and reduce the problem to the analysis of the associated cubic-nonlinear equation. We show that the solution in the form of a TE-polarized electromagnetic wave exists and can be obtained by iterating a cubic-nonlinear integral equation. We derive the associated dispersion equation and prove that it has a root that determines this solution.

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found.

Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found.

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